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started stub for A-model
What is called the A-model is the 2-dimensional topological conformal field theory corresponding to the Calabi-Yau category called the Fukaya category? of a symplectic manifold X.
Well that is a posteriori in the cases in which you develop 20 years of theory successfully. Topological A model is certain N=2 superconformal field theory. In SOME cases one can construct some version of Fukaya category (for different kinds of manifolds dfifferent definitions, for some there is none so far, while SCFT still exists). The equivalence between such infinity category approach and SCFT approach in some cases has been established only by around 2003 by Kontsevich and then by Cosetllo (Costello had first errors, and then corrected).
Second and more important, if you work with A-infinity category abstractly you do not know if this one came from A-model or from B-model or for anything else. Only if you have background like symplectic manifold or alike then you can talk about A-model.
formal physics
There is theoretical physics, there is mathematical physics, but “formal physics” I heard only from you. Is this a next proposed level of alienation of some branch of theoretical physics ?
Well that is a posteriori in the cases in which you develop 20 years of theory successfully.
Yes! And I do want the entry to use these 20 years of insights. .-)
Topological A model is certain N=2 superconformal field theory
No, it’s not an SCFT. It is obtained from the standard sigma-model SCFT by applying a “topological twi” as they say.
No, it’s not an SCFT. It is obtained from the standard sigma-model SCFT by applying a “topological twist” as they say.
this remark should be expanded and merged at the beginning of mirror symmetry (not feeling expert enough to do it myself)
but notice: at the beginning of the entry mirror symmetry it is correct to speak of SCTFs.
So first people wrote down the superstrring on CY targets and noticed (it is very easy to see in the vertex operator algebra, actually) that the worldsheet theories are the same for what are now called mirror partner CYs.
Then in order to study this duality in more detail, Witten introduced the two topological twists of the SCFT coming from a sigma-model on a CY, and showed that when switching between mirror partners now these two topological QFTs are mno longer invariant, but are exchanged against each other.
Then Kontsevich conjectured that this equivalence between the A-model and the B-model should go along with an equivalence of the Fukaya-category and the derived category associated with the CY. Then as time passed, people understood more and more why this should be the case: these categories are the categories of branes for these two theories.
With the Kontsevich-Costello work finally this is fully clarified: the A_oo-enhancement of the derived category of the CY compleztely encodes a TCFT on all worldsheet genera and thus really (for the first time) fully defines “the B-model” in a rigorous sense. And with the cobordism-hypothesis-style theorem that those TCFTs are entirely equivalent to CY A_oo categories, this then also explains why and how exactly the original homological mirror symmetry conjecture corresponds to an equiavence of two quantum field theories.
Oh dear, now I really have to run…
Urs,
I’m not sure I understand this: at the beginning of mirror symmetry is written “Mirror symmetry between two Calabi–Yau varieties and is a pair of isomorphisms between the N=2 superconformal field theories called A-model and the B-model attached to and .” so, from this sentence it seems A-model and B-model are SCFTs.
I agree with all the rest of your post
created stub for Fukaya category
No, it’s not an SCFT. It is obtained from the standard sigma-model SCFT by applying a “topological twi” as they say.
It is. The original paper of Vafa emphasises that the mirror symmetry respects the N=1 supersymmetry but not the remaining, so it is an equivalence of N=1 SCFTs but each of them is N=2. Calabi Yau A-infinity category itself is not a TQFT but it gives sufficient data to construct a TQFT, similarly a Calabi Yau manifold is not a TQFT but just gievs a data sufficient to construct a TQFT.
All you wrote in 5 is standard story. But the difference between N=1 and N=2 is basically in nonobservable plus or minus 1 eigenvalue of U(1) symmetry which is not of physical significance.
With the Kontsevich-Costello work finally this is fully clarified: the A_oo-enhancement of the derived category of the CY compleztely encodes a TCFT on all worldsheet genera and thus really (for the first time) fully defines “the B-model” in a rigorous sense.
Kontsevich expected that from the beginning in 1994, but could not complete the construction of SCFT out of A-infinity category up to 2003/2004.
I do not know how to address in the present wording but A-model should not be defined for Calabi-Yau but much more generally, roughly for closed symplectic manifolds (but not only them). Only the mirror symmetry needs that we restrict to something like Calabi-Yau, not the A-model and its manifestations like quantum cohomology, Floer homology etc.
But that’s what it does say at the beginning of A-model! (?)
No, the present A-model entry says that the A-infinity category considered for A-model for is Calabi-Yau what means that you silently assumed that the first Chern class is zero. The analogue of the Fukaya category for general closed symplectic manifold will be an A-infinity category which is smooth and proper but the conditions for Serre functor will not be generally satisfied (cf. e.g. arXiv/0806.0107 page 55). In complex dimension 3, first Chern class = 0 is basically the CY condition. So the page implictly restricts the whole theory to the CY case.
Ah, I see. Hm, but that’s a bti weird then, because without the category being CY, there is no TCFT associated with it.
A priori that is OK, as you have A-infinity categories in similar way also in some situations over other fields, not ocmplex numbers. On the other hand, I agree that we should better understand this “assymetry”.
I don’t think this is completely true. There is still the question of extending the TCFT to the DM moduli space.
So where is this discussed?
I learned this stuff by talking to Teleman and to Pantev, and from a talk of Kontsevich in 2008. But again, I don’t think any of this is really written up yet.
There are two sentences about this in the Acknowledgements section of Costello’s CY categories paper: “I benefitted greatly from an inspiring talk of M. Kontsevich at the Hodge centenary conference in 2003. In this talk he described several results related to those in this paper; in particular, he sketched a different construction of a TCFT structure on the Hochschild chains of an A-infinity algebra, and also an extension of this to the Deligne-Mumford spaces when the Hodge to de Rham spectral sequence degenerates.” Take a look at section 1.6 of v1 of that paper on the arxiv for more elaboration: http://arxiv.org/pdf/math/0412149v1 (this section was removed from the later versions for some reason). Some more related discussion on mathoverflow as well: http://mathoverflow.net/questions/8692/higher-genus-closed-string-b-model.
Probably Costello’s paper “The Gromov-Witten potential associated to a TCFT” will say a bit more. You might also be able to find a few words about this in Katzarkov-Kontsevich-Pantev.
Oh yeah, I just remembered, it should be in this paper of Kontsevich-Soibelman http://arxiv.org/abs/math/0606241.
Thanks, Kevin!
i am somewhat busy with a dozen other tasks. If you could store these references and comments you just gave into the relevant nLab entry for the moment, then I try to look into this again when I have a spare minute. Thanks!
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